Physics 176: Midterm Exam Solutions
March, 2011
Most of the following answers are more detailed than what was necessary to get full credit. I hope the details
will help you improve your problem solving skills and understand the physics better.
Problems That Require Writing
1. Black holes are simple pure objects in that, no matter what kind of matter collapsed to form the black
hole, only three numbers are needed to define its macrostate: the black hole’s mass
M
, its electrical
charge
Q
, and its angular momentum
L
. For an electricallyneutral nonrotating black hole (
Q
= 0,
L
=
0
), the black hole’s entropy
S
depends only on
M
and is given by
S
=
8
π
2
kG
hc
M
2
(1)
where
k
is Boltzmann’s constant,
G
is the gravitational constant,
h
is Planck’s constant, and
c
is the
speed of light.
(a)
(5 points)
Given that the energy of a black hole is its relativistic rest mass
U
=
Mc
2
, derive a
formula for the temperature
T
of a black hole in terms of its mass
M
. Does decreasing the mass
of a black hole make it hotter or colder?
Answer:
The temperature can be deduced from the dependence of the entropy
S
(
U
) on the
system’s energy
U
using the standard formula. We have:
1
T
=
∂S
∂U
(2)
=
∂S
∂M
×
dM
dU
(3)
=
8
π
2
kG
hc
·
2
M
×
1
c
2
(4)
=
16
π
2
kG
hc
3
M,
(5)
so the answer is
T
=
hc
3
16
π
2
GkM
∝
M

1
.
(6)
Note how the entropy and temperature of a black hole involve fundamental constants from four
different areas of physics: thermal physics (
k
), gravity (
G
), quantum mechanics (
h
), and special
relativity (
c
). This makes black holes intriguing indeed.
Since
T
∝
M

1
and mass is always a positive quantity (any material object resists acceleration),
the temperature must also be positive for a black hole. Black holes are exotic objects but they
can’t have a negative temperature like a paramagnet.
Because
T
∝
M

1
, the smaller the mass of a black hole, the hotter it is.
A sufficiently small
black hole (about the mass of the Earth’s moon or less, 10
22
kg) will become hotter than 3 K
the temperature of the cosmic blackbody radiation.
We would then expect energy to transfer
from the black hole to the surrounding photon gas if the black hole could transfer energy. This
has been predicted to be possible by the physicist Stephen Hawking, who heuristically applied
relativistic quantum mechanics to Einstein’s classical gravity theory to predict that all black holes
emit light radiation from their event horizon (called “Hawking radiation”) and so can evaporate
away into light if hotter than their surroundings. Further, the smaller the mass of a black hole,
the faster it loses mass by Hawking radiation. The tiny small mass black holes predicted to occur
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
in the Large Hadron Collider (based on the assumption that there are more than three spatial
dimensions) are super hot and disappear within 10

10
s of their appearance, in a burst of gamma
rays which would be the signature for their appearance. You can learn more about Eq. (6), called
the Hawking radiation temperature, from the Wikipedia article with title “Hawking Radiation”.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Behringer
 Physics, Thermodynamics, Black Holes, Otto cycle

Click to edit the document details